886 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



and (4.10) becomes a good approximation to (4.7). Just how small Po 

 has to be depends upon the signal bandwidth, fb , and the characteristics 

 of the transducer. 



2. For the case of FM and a flat signal band, the second of equations 

 (5.3) shows that even if Po is small, the difference \po — \pr approaches co 

 as I T I approaches =» . To justify the use of (4.10) in this case it is neces- 

 sary to take into account the behavior of g{t), the response of the trans- 

 ducer to the unit impulse b{t). For example, if the duration of g{x) in 

 (4.7) is so brief that g{x) becomes negUgibly small before —\}/o-\- ypx be- 

 comes appreciably different from zero (which may be achieved by mak- 

 ing Po small enough) then (4.10) is a good approximation to (4.7). 



3. When the attenuation, a, and phase shift, /S, are given for any 

 particular transducer, the corresponding g{t) may be obtained from (1.2). 

 Once g{t) and ypo — ^t are known, the conditions under which exp ( — i/'o + 

 yp^ may be replaced by unity in (4.7) and 0{Rv) terms neglected in (3.2) 

 may be determined by direct examination of the integrals. 



As might be expected, the third order modulation results are quite 

 complicated. The third order modulation term in (3.2) is 



Qii f ^/' f drwM')wM">M"') 



o!4 J — 00 J— 00 , . 



(4.11) 



cospxe-^°+^^(2'' - 1)(/" - 1)(/"' - 1) 



/oo 

 dxgix) 

 -00 



where /'" = f — f — J" and z = exp ( — i2Trx) . When i/'o is small this is 

 approximately 



^ f df r drwM')wM")^M'")[H' + K'] (4.12) 

 3! lb J-oo ^-00 



where 



H = 7n{n + mXn + mif) + m{f - f - f") 



- ni{f - n - m{f - n - m(0) - m(f' + /"), 



m{f) = Gf + (?_/, n(/) = Bj - 5_,, 

 and K is an expression obtained from H by replacing n by m. 



V MISCELLANEOUS COMMENTS 



Here we make some miscellaneous comments related to the foregoing 

 results. 



